# On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

## TL;DR

This paper proves the global convergence of certain parallel cyclic Jacobi methods for symmetric matrices of order 4, showing they consistently reduce the off-diagonal norm, and discusses the speed variability depending on matrix properties.

## Contribution

It establishes the global convergence of specific parallel Jacobi strategies for 4x4 symmetric matrices, a result not previously confirmed for these methods.

## Key findings

- The inequality S(A^{[2]}) ≤ (1 - 10^{-5}) S(A) holds for all symmetric 4x4 matrices after two cycles.
- The method's convergence is guaranteed under all fully parallel strategies considered.
- There exist matrices where the first cycle does not significantly reduce the off-diagonal norm, indicating variability in convergence speed.

## Abstract

The paper analyzes special cyclic Jacobi methods for symmetric matrices of order $4$. Only those cyclic pivot strategies that enable full parallelization of the method are considered. These strategies, unlike the serial pivot strategies, can force the method to be very slow or very fast within one cycle, depending on the underlying matrix. Hence, for the global convergence proof one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix $A$ of order~$4$ the inequality $S(A^{[2]})\leq(1-10^{-5})S(A)$ holds, where $A^{[2]}$ results from $A$ by applying two cycles of a particular parallel method. Here $S(A)$ stands for the Frobenius norm of the strictly upper-triangular part of $A$. The result holds for two special parallel strategies and implies the global convergence of the method under all possible fully parallel strategies. It is also proved that for every $\epsilon>0$ and $n\geq4$ there exist a symmetric matrix $A(\epsilon)$ of order $n$ and a cyclic strategy, such that upon completion of the first cycle of the appropriate Jacobi method the inequality $S(A^{[1]})> (1-\epsilon)S(A(\epsilon))$ holds.